Considering Copositivity Locally

Let $A$ be an element of the copositive cone $\mathcal{COP}^n$. A zero $\mathbf{u}$ of $A$ is a nonnegative vector whose elements sum up to one and such that $\mathbf{u}^TA\mathbf{u} = 0$. The support of $\mathbf{u}$ is the index set $\mathrm{supp}\mathbf{u} \subset \{1,\dots,n\}$ corresponding to the nonzero entries of $\mathbf{u}$. A zero $\mathbf{u}$ of $A$ is called minimal if there does not exist another zero $\mathbf{v}$ of $A$ such that its support $\mathrm{supp}\mathbf{v}$ is a strict subset of $\mathrm{supp}\mathbf{u}$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $\mathcal{K}^A$ of real symmetric $n \times n$ matrices $B$ such that there exists $\delta > 0$ satisfying $A + \delta B \in \mathcal{COP}^n$. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in $\mathcal{COP}^n$, and of the minimal exposed face of $A$ in $\mathcal{COP}^n$, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of $\mathcal{COP}^n$ by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $\delta > 0$ we have $A - \delta C \not\in \mathcal{COP}^n$.





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